6 CHRISTOPH HAUERT, NINA HAIDEN, AND KARL SIGMUND
(a) G1 G2
G3
G4
(b) G1 G2
G3
G4
Figure 3. Replicator dynamics for public goods games with pun-ishment and reputation: (a) for group sizes of N = 5 and (b) N = 10 players. Introducing reputation results in a bi-stable sit-uation. Depending on the initial configuration, the systems ends either in the social equilibrium G1 or the asocial G3. Instead of Q and the line of fixed points along G1G4 there appears an in-terior fixed point which essentially determines the size of the two basins of attraction. For increasing group sizesN the fixed point approachesG4and thereby increases the basin of attraction ofG1. Parameters: r= 1.5, c= 4, γ= 1, β= 2, µ= 0.2.
the public goods game, but can instead turn to some autark activity yielding an average payoff σ which is unaffected by the other players. To fix ideas, imagine that within the large population, random samples ofN players are asked whether they wish to engage in a public goods game or prefer the autark activity. We shall consider only three strategies: the cooperators and the defectors, who opt for the public goods game, with the intention either to contribute or to exploit, and the loners, who prefer not to join the group of public goods players anyway. These three strategies are fixed in advance, and do not depend on the size or composition of the group playing the public goods game. But it can happen, of course, that in a sample only a single cooperator or defector is willing to engage in the public goods game. In this case, the game will not take place and the players must go for autarky.
We denote by x, y and z the frequencies of cooperators, defectors and loners, respectively, and byPx, Py andPz their average payoff. Clearly
Pz=σ (18)
where we shall always assume that
0< σ <(r−1)c. (19) The frequency of cooperators among the players actually willing to join a public goods group is
f = x
x+y. (20)
The payoff for a defector in a group of S players, of which m are cooperators, is mrc/S. Both m andS are random variables. For any givenS the average payoff
THE DYNAMICS OF PUBLIC GOODS 7
for a defector is specified by
S−1
and therefore the average payoff for a defector who is not the only member of the group is Altogether this yields an average payoff for a defector
Py =σzN−1+f rc(1− 1−zN
N(1−z)). (23)
Given that there areS−1 co-players in the group, switching from being a defector to being a cooperator yieldsc(1−Sr). For the cooperator’s average payoffPx one therefore obtains
an expression which depends neither on f nor on σ. The average payoff in the population ¯P =xPx+yPy+zPz is given by
P¯=σ−[(1−z)σ−c(r−1)x](1−zN−1). (27) Due to assumption (19), the three strategies form a rock-scissors-paper cycle: if most players cooperate, it is best to defect; if most players defect, it is best to abstain from the public goods game; and if most players are loners, it is best to cooperate. It is only this third statement which is non-intuitive. But if the frequency of loners is high, then most groups are small, and among mostly small groups, cooperation can be a better option than defection. Indeed, in spite of the fact that within every group, defectors do better than cooperators (by economising their own contribution), it can happen that across all groups, cooperators do better, on average, than defectors. This is an instance of Simpson’s paradox.
In order to study the dynamics, it is convenient to effectuate a change in vari-ables, and consider, instead of (x, y, z)∈S3, the two variables (f, z)∈[0,1]×[0,1]. Dividing the right hand sides of the previous two equations by the positive factor f(1−f)z(1−z)(1−zN−1), which corresponds to a change in velocity and does not affect the orbits, one obtains
f˙= −cF(z)
z(1−z)(1−zN−1) (30)
8 CHRISTOPH HAUERT, NINA HAIDEN, AND KARL SIGMUND
0.0 0.2 0.4 0.6 0.8 1.0
f
0.0 0.2 0.4 0.6 0.8 1.0
z R
Figure 4. Replicator dynamics for optional public goods games:
in order to prove that forr >2 all orbits are closed (see figure 5b), the replicator equation is rewritten as a Hamiltonian system on ]0,1[×]0,1[ through an appropriate change of variablesf =x+yx .
and
˙
z=σ−c(r−1)f
f(1−f) (31)
which is a Hamiltonian system for (f, z) in [0,1]×[0,1]. Forr≤2 one has always f →0 because F(z) is positive on [0,1[. For r >2 there exists a unique zero ˆz of F(z) in ]0,1[. This follows from the fact that G(z) := (1−z)F(z) (which has the same zeros asF(z) in ]0,1[) satisfiesG(0)>0, is negative forz6= 1 close to 1 and has a second derivative
G00(z) =zN−3(N−1)[(N−2)(r−1)−z(N r−N−r)] (32) which changes sign only once in ]0,1[. In this case all orbits in ]0,1[×]0,1[ are closed orbits surrounding (c(r−1)σ ,ˆz), see figure 4.
Translating this into the replicator dynamics onS3, one sees that forr≤2 the point (0,0,1) (loners only) is a homoclinic rest point (see figure 5a), whereas for r >2, all orbits in intS3 are closed orbits surrounding (ˆx,y,ˆ z) whereˆ
ˆ
x = σ
c(r−1)(1−z)ˆ (33)
ˆ
y = (1− σ
c(r−1))(1−z)ˆ (34) (see figure 5b). We note that by increasing the sample sizeN, the equilibrium value ˆ
zincreases: there will be less and less willingness to participate in the public goods, but f and hence the ratio between cooperators and defectors remains unchanged.
Increasing the loner’s payoffσleaves the loner’s frequency unchanged, and increases the equilibrium value ˆxof cooperators. Increasing the multiplication factorr(the
’interest rate’ of the public good) results in a larger equilibrium value ˆyof defectors.
It is easy to see that the time-averages for the payoff values Px, Py and Pz must all be equal, and hence equal to σ. Thus, in spite of endless oscillations in the population, no type does better, on average, than the loners. The public goods is a
THE DYNAMICS OF PUBLIC GOODS 9
(a) ez ey
ex
(b) ez ey
ex
Q
Figure 5. Replicator dynamics for optional public goods games in S3: (a) for r < 2 and (b) for r > 2. The cyclic dominance of the three strategies is reflected in the heteroclinic cycle along the boundary of the simplex S3. (a) For low multiplication factors, intS3 consists of homoclinic orbits only. Except for brief inter-mittent bursts of cooperation due to random shocks, the system always remains inez. (b) In contrast, for higherran interior fixed pointQappears surrounded by closed orbits. This results in peri-odic oscillations of cooperators, defectors and loners. Parameters:
(a)N = 5, r= 1.8, c= 1, σ= 0.5, (b)N = 5, r= 3, c= 1, σ= 1.
tempting option, but it always gets undermined by defection. On the other hand, the option of dropping out of the game leads to ever recurrent bursts of cooperation.
4. Discussion. In this paper we have postulated that the frequencies of strategies